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Article Contents

# The asymptotic behavior of solutions of a semilinear parabolic equation

• We study the long-time behavior of solutions of the Cauchy problem

$u_t=\Delta u - (u^q)_y- u^p, \quad p, q >1,$

defined in the domain $Q=\{ (x, t): x=(x, y) \in \mathbf{R}^{N-1} \times \mathbf{R}, t >0 \}$ with nonnegative initial data in $L^1( \mathbf{R}^N)$. We completely classify the asymptotic profiles of solutions as $t \to \infty$ according to the parameters $p$ and $q$. We use rescaling transformations and a priori estimates.

Mathematics Subject Classification: 35B30, 35B40, 35K15.

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